In this paper we discuss a functional approach to obtain a lattice-like structure in d -algebras, and obtain an exact analog of De Morgan law and some other properties. Y. Imai and K. Iséki introduced two classes of abstract algebras: BCK-algebras and BCI -algebras ( [8 , 9] ). BCK -algebras have some connections with other areas: D. Mundici [13] proved that MV -algebras are categorically equivalent to bounded commutative BCK -algebras, and J. Meng [11] proved that implicative commutative semigroups are equivalent to a class of BCK -algebras. It is well known that bounded commutative BCK -algebras, D -posets and MV -algebras are logically equivalent each other (see [4, p. 420]). We refer useful textbooks for BCK / BCI -algebra to [4 , 6 , 7 , 12 , 17] . J. Neggers and H. S. Kim ( [14] ) introduced the notion of d-algebras which is a useful generalization of BCK -algebras, and then investigated several relations between d -algebras and BCK -algebras as well as several other relations between d -algebras and oriented digraphs. J. S. Han et al. ( [5] ) defined a variety of special d -algebras, such as strong d -algebras, (weakly) selective d -algebras and others. The main assertion is that the squared algebra ( X ; □, 0) of a d -algebra is a d -algebra if and only if the root ( X ; ∗, 0) of the squared algebra ( X ; □, 0) is a strong d -algebra. Recently, the present author with H. S. Kim and J. Neggers ( [10] ) explored properties of the set of d -units of a d -algebra. It was noted that many d -algebras are weakly associative, and the existence of non-weakly associative d / BCK -algebras was demonstrated. Moreover, they discussed the notions of a d -integral domain and a left-injectivity in d / BCK -algebras. We refer to [1 , 2 , 15 , 16] for more information on d -algebras. In this paper we discuss a functional approach to obtain a lattice-like structure in d -algebras, and obtain an exact analog of De Morgan law and some other properties. An ( ordinary ) d - algebra ( [14] ) is a non-empty set X with a constant 0 and a binary operation “ ∗ ” satisfying the following axioms: A BCK -algebra is a d -algebra X satisfying the following additional axioms: Example 2.1 ( [14] ). Consider the real numbers R , and suppose that ( R ; ∗, e ) has the multiplication x ∗ y = ( x − y )( x − e ) + e Then x ∗ x = e ; e ∗ x = e ; x ∗ y = y ∗ x = e yields ( x − y )( x − e ) = 0, ( y − x )( y − e ) = e and x = y or x = e = y , i.e., x = y , i.e., ( R ; ∗, e ) is a d -algebra. Let ( X , ∗, 0) be a d -algebra. A map ϕ : X → X is said to be order reversing if x ∗ y = 0 then ϕ ( y ) ∗ ϕ ( x ) = 0 for all x , y ∈ X ; self-inverse if ϕ ( ϕ ( x )) = x for all x ∈ X ; an anti-homomorphism if ϕ ( x ∗ y ) = ϕ ( y ) ∗ ϕ ( x ) = 0 for all x , y ∈ X ; a homomorphims if ϕ ( x ∗ y ) = ϕ ( x ) ∗ ϕ ( y ) for all x , y ∈ X . Example 3.1. Consider X := {0, a , 1} with Then ( X ; ∗, 0) is a d -algebra. If we define a map ϕ : X → X by ϕ (0) = 1, ϕ ( a ) = a and ϕ (1) = 0, then it is easy to see that ϕ is both self-inverse and order reversing, but it is not an anti-homomorphism, since ϕ ( a ∗1) = ϕ (0) = 1 and ϕ (1)∗ ϕ ( a ) = 0∗ a = 0. Moreover, it is not a homomorphism, since ϕ (0 ∗ a ) = ϕ (0) = 1 ≠ a = 1 ∗ a = ϕ (0) ∗ ϕ ( a ). Proposition 3.2. Let ( X , ∗, 0) be a d-algebra . If ϕ : X → X is a (anti-) homomorphism, then ϕ (0) = 0. Proof . Since X is a d -algebra, by ( D 1), we obtain ϕ (0) = ϕ ( x ∗ x ) = ϕ ( x ) ∗ ϕ ( x ) = 0. Proposition 3.3. If ( X , ∗, 0) i s a d-algebra, then every anti- homomorphism is order reversing . Proof . Let ϕ : X → X be an anti-homomorphism. If we assume that x ∗ y = 0, then ϕ ( y ) ∗ ϕ ( x ) = ϕ ( x ∗ y ) = ϕ (0) = 0 by Proposition 3.2. This proves the proposition. Remark. The converse of Proposition 3.3 need not be true in general. In Example 3.1, the mapping ϕ is an order reversing, but not an anti-homomorphism. Let ( X , ∗, 0) be a d -algebra and let ϕ : X → X be a map. We denote by 1 := ϕ (0). Proposition 3.4. Let ( X , ∗, 0) be a d-algebra and let ϕ : X → X be both order reversing and self-inverse . Then ( X , ∗, 0) is bounded. Proof . Given x ∈ X , we have Let ( X , ∗, 0) be a d -algebra. We define a relation “≤” on X by x ≤ y if and only if x ∗ y = 0 for all x , y ∈ X . Note that the relation ≤ need not be a partial order on X . We define a relation “∧ on X by x ∧ y := x ∗ ( x ∗ y )) for all x , y ∈ X . Proposition 3.5. Let ( X , ∗, 0) be a d-algebra . If ϕ : X → X is self-inverse , then ϕ (1) = 0. Proof . It follows from ϕ is self-inverse that 0 = ϕ ( ϕ (0)) = ϕ (1). Theorem 3.6. Let ( X , ∗, 0) be a d-algebra and let ϕ : X → X be a self-inverse map . If we define x ∨ y := ϕ [ ϕ ( y ) ∧ ϕ ( x )], then ϕ ( x ∧ y ) = ϕ ( y ) ∨ ϕ ( x ) for all x , y ∈ X . Proof . Given x , y ∈ X , we have Theorem 3.6 shows that the first De Morgan’s law implies the analog of the second De Morgan’s law and conversely, since x ∨ y ≠ y ∨ x in general. Moreover, it follows that x ∧ y = ϕ ( ϕ ( x ∧ y )) = ϕ [ ϕ ( y ) ∨ ϕ ( x )] for all x , y ∈ X . Theorem 3.7. Let ( X , ∗, 0) be a d-algebra with for all x ∈ X . If ϕ : X → X is a self-inverse map , then x ∨ x = x and x ∧ x = x for all x ∈ X . Proof . (i). Given x ∈ X , we have (ii). x ∧ x = x ∗ ( x ∗ x ) = x ∗ 0 = x . Proposition 3.8. Let ( X , ∗, 0) be a d-algebra with for all x , y , z ∈ X . Then x ∧ y ≤ x and x ∧ y ≤ y for all x , y ∈ X . Proof . (i). Given x , y ∈ X , by applying (2), we obtain (ii). Given x , y ∈ X , we have ( x ∧ y )∗ y = ( x ∗( x ∗ y ))∗ y = ( x ∗ y )∗( x ∗ y ) = 0. Theorem 3.9. Let ( X , ∗, 0) be a d-algebra with the condition ( 2 ). If ϕ : X → X is a self-inverse anti-homomorphism , then x ∗ ( x ∨ y ) = 1 and y ∗ ( x ∨ y ) = 1 f or all x , y ∈ X . Proof . (i). Since ϕ : X → X is a self-inverse anti-homomorphism, for all x , y ∈ X , we obtain and Whether such functions exists or not depends on the special properties of the d -algebras. BCK -algebras have the partial order structure, but d -algebras have no such a structure and so we need to seek another conditions for obtaining the analog of structures in d -algebras. This kind of functional approach can be connected with mirror d -algebras discussed in [3] in a new direction.